Third harmonic characterization of antiferromagnetic heterostructures

Electrical switching of antiferromagnets is an exciting recent development in spintronics, which promises active antiferromagnetic devices with high speed and low energy cost. In this emerging field, there is an active debate about the mechanisms of current-driven switching of antiferromagnets. For heavy-metal/ferromagnet systems, harmonic characterization is a powerful tool to quantify current-induced spin-orbit torques and spin Seebeck effect and elucidate current-induced switching. However, harmonic measurement of spin-orbit torques has never been verified in antiferromagnetic heterostructures. Here, we report harmonic measurements in Pt/α-Fe2O3 bilayers, which are explained by our modeling of higher-order harmonic voltages. As compared with ferromagnetic heterostructures where all current-induced effects appear in the second harmonic signals, the damping-like torque and thermally-induced magnetoelastic effect contributions in Pt/α-Fe2O3 emerge in the third harmonic voltage. Our results provide a new path to probe the current-induced magnetization dynamics in antiferromagnets, promoting the application of antiferromagnetic spintronic devices.


2) In-plane angular-dependent Harmonic voltages in Pt/-Fe 2 O 3 (0001)
When an AC current sin is applied, due to spin-orbit torque (SOT) and thermoelectric effects, the measured transverse voltage ∑  sin where  is called n th harmonic voltage. Here we show the harmonic voltages in Pt/-Fe2O3 (0001) in an inplane magnetic field.
Current-induced spin-orbit torques and magnetoelastic effect drive the antiferromagnetic (AFM) moment slightly off the equilibrium orientation, which change the transverse resistance of Pt/-Fe2O3 (0001) bilayers. Thus, where is the equilibrium transverse resistance at the limit of 0, and, | , (S1-2)  | , (S1-3)  | , (S1-4)  | , (S1-5) where  ,  and  are the first, second, and third harmonic voltages, which are proportional to , and , respectively.  Next, we show the torque balance equations, which are similar to those for ferromagnets, where is the effective magnetic field that includes the following contributions:  External magnetic field ,  Exchange field ,  Effective field of Dzyaloshinskii-Moriya (DM) interaction ,  Anisotropy field . 1 ( is the easy-plane anisotropy. is the effective easy axis anisotropy due to magnetoelastic (ME) effect).
We assume the external magnetic field is large enough so that the -Fe2O3 film is in single domain state and we ignore the small in-plane tri-axial anisotropy. The field-like torque and the damping-like torque are, where is the unit vector of spin polarization in Pt along , and are the effective fields of field-like and damping-like torques, respectively. Here where ∇ is the current-induced temperature gradient along z. 1 At 0 with the external magnetic field applied in the xy plane, the equilibrium orientations of the sublattice and as well as the Néel order and net magnetization can be described by the following polar and azimuthal angles (see Fig. S2 For field-like torque, For damping-like torque, As we can see from Eqs. S6-1 and S6-2, Δ Δ and Δ Δ . This is because the opposite sublattices act symmetrically on which tilts them out-of-plane in opposite directions. To check the explicit solution of Eqs. (S5)-(S7), we numerically solve the coupled Landau- where is gyromagnetic ratio and is Gilbert damping constant. Figure S3 shows the simulated results, which agree well with the explicit solution using Eqs. (S5)-(S7).  (0001) in an in-plane magnetic field , which is called the negative transverse spin Hall magnetoresistance (TSMR). 3,4 Then, the derivative resistance terms in Eq. (S1) can be derived as, Here we neglect higher order terms. Thus,  sin 2 sin 2 (S11) For field-like torque term,  | cos 2 cos (S12-1) For damping-like torque, For magnetoelastic effect,

3) Magnetoelastic effect at different temperatures
As temperature varies, several important parameters such as thermal conductivity, heat capacity and thermal expansion coefficient change considerably, which may impact the Néel order of the antiferromagnets. Table S1 lists all key parameters that are related to magnetoelastic effect.
Using COMSOL, we simulate the current-induced anisotropic compressive stress Δσ at different temperatures (see Fig. S4). Table S1. Parameters used in the simulation of magnetoelastic effect in a Pt(5 nm)/-Fe 2 O 3 (30 nm) bilayer. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The conductivity of Pt is measured in our sample.  Figure S5 shows the different harmonic voltage components at different temperatures. We find that the spin Seebeck coefficient and the effective field that is attributed to field-like torque at multiple domain state have a weak temperature dependence. The conventional field-like torque effective field has a strong temperature dependence as reported in previous research; [27][28][29] however, the detailed mechanism is under debate, and future study is needed to reveal the underlying physics.

4) More harmonic measurement results
The  contribution related to the change of Pt resistivity also decreases with temperature, which is due to the decrease of ΔT as shown in Fig. S4b. We also make similar harmonic measurements for the samples with different -Fe2O3 thicknesses. Figure S6 shows the results of Pt(5 nm)/-Fe2O3 bilayers with the thicknesses of -  Figure S7 shows the different harmonic voltage components as a function of current. Based on the model we build using Eqs. S11-S18, it is expected that: These five components are plotted as a function of current in Figs. S7a-S7d and fitted by the corresponding polynomial functions with the expected current dependence.
Here, is the real (imaginary) part of spin mixing conductance which determines the magnitude of DL(FL)-SOT. In HM/metallic-ferromagnet systems, ≫ , which leads to a much larger compared with . This is confirmed by both experiments and first-principle calculations. 30,31 However, in HM/magnetic-insulator systems which receive much attention recently, this is not always the case. In Pt/Y3Fe5O12 (YIG) and Pt/EuS bilayers, 32-34 much larger than has been reported. Large in our Pt/-Fe2O3 might also be related to the insulating property of -Fe2O3. To date, however, there are only few works that calculated the spin mixing conductance in HM/magnetic-insulator bilayers, 35 but failed to match the recent experimental results shown above. Further research in HM/magnetic-insulator systems is required address this question.

6) Thermal effect on the magnetic parameters
-Fe2O3 has a very high Néel temperature ~950 K. Thus, its magnetic parameters, for example the sublattice magnetization, barely depend on temperature in our measurement temperature range.
This has been confirmed in our previous work. 8